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<h1>Weighted analytic center of a set of linear inequalities</h1>
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<pre class="codeinput">
<span class="comment">% Jo&Atilde;&laquo;lle Skaf - 04/29/08</span>
<span class="comment">%</span>
<span class="comment">% The weighted analytic center of a set of linear inequalities:</span>
<span class="comment">%           a_i^Tx &lt;= b_i   i=1,...,m,</span>
<span class="comment">% is the solution of the unconstrained minimization problem</span>
<span class="comment">%           minimize    -sum_{i=1}^m w_i*log(b_i-a_i^Tx),</span>
<span class="comment">% where w_i&gt;0</span>

<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>, 0);
rand(<span class="string">'state'</span>, 0);
n = 10;
m = 50;
tmp = randn(n,1);
A = randn(m,n);
b = A*tmp + 2*rand(m,1);
w = rand(m,1);

<span class="comment">% Analytic center</span>
cvx_begin
    variable <span class="string">x(n)</span>
    minimize <span class="string">-sum(w.*log(b-A*x))</span>
cvx_end

disp(<span class="string">'The weighted analytic center of the set of linear inequalities is: '</span>);
disp(x);
</pre>
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<pre class="codeoutput">
 
Calling Mosek 9.1.9: 150 variables, 60 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 60              
  Cones                  : 50              
  Scalar variables       : 150             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 60              
  Cones                  : 50              
  Scalar variables       : 150             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 10
Optimizer  - Cones                  : 50
Optimizer  - Scalar variables       : 150               conic                  : 150             
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 55                after factor           : 55              
Factor     - dense dim.             : 0                 flops                  : 1.14e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   5.8e+00  3.2e+00  8.3e+01  0.00e+00   6.028258137e+01   -2.204216600e+01  1.0e+00  0.00  
1   1.9e+00  1.0e+00  1.8e+01  6.36e-01   2.402764877e+01   -7.490705411e+00  3.3e-01  0.01  
2   6.6e-01  3.6e-01  4.1e+00  7.86e-01   7.484274349e+00   -4.700807210e+00  1.1e-01  0.01  
3   4.0e-01  2.2e-01  2.0e+00  8.15e-01   2.978822508e+00   -4.803366948e+00  6.8e-02  0.01  
4   1.5e-01  8.2e-02  5.3e-01  7.16e-01   -2.181211154e+00  -5.559357961e+00  2.6e-02  0.01  
5   4.9e-02  2.7e-02  1.1e-01  8.18e-01   -4.601865218e+00  -5.810948806e+00  8.5e-03  0.01  
6   1.1e-02  6.0e-03  1.2e-02  8.75e-01   -5.671320321e+00  -5.959202164e+00  1.9e-03  0.01  
7   1.8e-03  9.9e-04  8.3e-04  9.69e-01   -5.937967101e+00  -5.985922606e+00  3.1e-04  0.01  
8   3.2e-05  1.7e-05  2.0e-06  9.89e-01   -5.991594363e+00  -5.992445856e+00  5.5e-06  0.01  
9   1.1e-06  6.0e-07  1.3e-08  9.99e-01   -5.992505691e+00  -5.992535031e+00  1.9e-07  0.01  
10  1.2e-07  6.1e-08  4.2e-10  1.00e+00   -5.992537162e+00  -5.992540134e+00  1.9e-08  0.01  
11  4.1e-08  9.9e-09  2.8e-11  1.01e+00   -5.992540319e+00  -5.992540801e+00  3.3e-09  0.01  
12  2.9e-08  1.5e-09  1.6e-12  1.02e+00   -5.992540760e+00  -5.992540832e+00  5.0e-10  0.01  
13  2.9e-08  1.5e-09  1.6e-12  1.00e+00   -5.992540760e+00  -5.992540832e+00  5.0e-10  0.02  
14  2.9e-08  1.5e-09  1.6e-12  1.00e+00   -5.992540760e+00  -5.992540832e+00  5.0e-10  0.02  
Optimizer terminated. Time: 0.02    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -5.9925407602e+00   nrm: 2e+01    Viol.  con: 5e-08    var: 0e+00    cones: 7e-10  
  Dual.    obj: -5.9925408321e+00   nrm: 3e+00    Viol.  con: 0e+00    var: 1e-09    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 15        time: 0.02    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.99254
 
The weighted analytic center of the set of linear inequalities is: 
   -0.5100
   -1.4794
    0.3397
    0.1944
   -1.0444
    1.1956
    1.3927
   -0.2815
    0.2863
    0.3779

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